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Unformatted text preview: n (with units)?
T = 2π sec
(b) What is the fundamental frequency ω of the function (with units)?
f = 1/T = 1/2π
ω = 2πf = 1 rad/sec
(c) Is y(t) an even or odd function?
odd
(c) Using Fourier series definition below, find the Fourier coefficients An and Bn for y(t) where: Hint 1: note that 1 ∫ t sin(nt) = n 2 t
sin( nt ) − cos(nt )
n To start, note that the average value of the y(t) over one full period is zero, and thus Ao = 0. More
generally, the function is an odd function, and thus An = 0 for all n. The Bn coefficients are found
as follows: €
2π
−1 π
Bn =
∫ (−t ) sin(nt)dt = π ∫ (t) sin(nt)dt
2π −π
−π Ⱥ 2 / n n even
Ⱥπ
−1 Ⱥ 1
t
2
= Ⱥ 2 sin(nt ) − cos( nt ) Ⱥ = cos(nπ ) = Ⱥ
Ⱥ −π n
π Ⱥ n
n
Ⱥ −2 / n n odd € Thus the Fourier series becomes:
2
1
y ( t ) = 2 sin(t ) − sin(2 t ) + sin(3t ) − sin( 4 t ) + ...
3
2
€
3 € € 2
or : Bn = (−1) n
n 30 pts 3. Consider the following Fourier series representation of a periodic function y(t):
y(t) = 8 + 12sin(2π t)  8sin(3π t) + 4sin(4π t)  sin(5π t) + …
(a) Determine the fundamental frequency (in units of Hz) for y(t).
Note: the true fundamental frequency should be π , since the higherorder frequencies are
multiples of π , but the π term is missing from the given series, so that 2π is the first
frequency term. This is an error in the exam, so either answer for ω is acceptable. Let's
u se ω = 2...
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This document was uploaded on 02/06/2014.
 Spring '14

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